Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 006, 13 pages      arXiv:1009.0408
Contribution to the Proceedings of the International Workshop “Recent Advances in Quantum Integrable Systems”

Coordinate Bethe Ansatz for Spin s XXX Model

Nicolas Crampé a, b, Eric Ragoucy c and Ludovic Alonzi c
a) Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France
b) CNRS, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France
c) LAPTh, CNRS and Université de Savoie, 9 chemin de Bellevue, BP 110, 74941, Annecy-Le-Vieux Cedex, France

Received September 06, 2010, in final form January 05, 2011; Published online January 12, 2011

We compute the eigenfunctions and eigenvalues of the periodic integrable spin s XXX model using the coordinate Bethe ansatz. To do so, we compute explicitly the Hamiltonian of the model. These results generalize what has been obtained for spin 1/2 and spin 1 chains.

Key words: coordinate Bethe ansatz; spin chains.

pdf (398 kb)   tex (18 kb)


  1. Heisenberg W., Zur Theorie des Ferromagnetismus, Z. Phys. 49 (1928), 619-636.
  2. Bethe H., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205-226.
  3. Kulish P.P., Sklyanin E.K., Quantum inverse scattering method and the Heisenberg ferromagnet, Phys. Lett. A 70 (1979), 461-463.
  4. Takhtajan L.A., Faddeev L.D., The quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979), 11-68.
  5. Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum Integrable Systems, Editor Mo-Lin Ge, Singapore, Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ, 1992, 63-97, hep-th/9211111.
  6. Vichirko V.I., Reshetikhin N.Yu., Excitation spectrum of the anisotropic generalization of an SU(3) magnet, Theoret. and Math. Phys. 56 (1983), 805-812.
    Reshetikhin N.Yu., A method of functional equations in the theory of exactly solvable quantum systems, Lett. Math. Phys. 7 (1983), 205-213.
    Reshetikhin N.Yu., The functional equation method in the theory of exactly soluble quantum systems, Sov. Phys. JETP 57 (1983), 691-696.
    Reshetikhin N.Yu., Integrable models of quantum one-dimensional magnets with O(n) and Sp(2k) symmetry, Theoret. and Math. Phys. 63 (1985), 555-569.
    Reshetikhin N.Yu., The spectrum of the transfer matrices connected with Kac-Moody algebras, Lett. Math. Phys. 14 (1987), 235-246.
  7. Kulish P.P., Reshetikhin N.Y., Sklyanin E.K., Yang-Baxter equation and representation theory. I, Lett. Math. Phys. 5 (1981), 393-403.
  8. Faddeev L.D., How algebraic Bethe ansatz works for integrable model, in Symétries Quantiques (Les Houches, 1995), Editors A. Connes, K. Gawedzki and J. Zinn-Justin, Les Houches Summerschool Proceedings, Vol. 64, North-Holland, Amsterdam, 1998, 149-219, hep-th/9605187.
  9. Lima-Santos A., Bethe ansätze for 19-vertex models, J. Phys. A: Math. Gen. 32 (1999), 1819-1839, hep-th/9807219.
  10. Takhtajan L.A., Introduction to algebraic Bethe ansatz, in Exactly Solvable Problems in Condensed Matter and Field Theory, Editors B.S. Shastry, S.S. Jha and V. Singh, Lecture Notes in Physics, Vol. 242, Springer, Berlin - Heidelberg, 1985, 175-220.
  11. Zamolodchikov A.B., Fateev V.A., A model factorized S-matrix and an integrable spin-1 Heisenberg ferromagnet, Soviet J. Nuclear Phys. 32 (1980), 298-303.
    Takhtajan L.A., The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins, Phys. Lett. A 87 (1982), 479-482.
    Babujian H., Exact solution of the isotropic Heisenberg chain with arbitrary spins: thermodynamics of the model, Nuclear Phys. B 215 (1983), 317-336.
  12. Gaudin M., La fonction d'onde de Bethe, Masson, Paris, 1983.
  13. Essler F.H.L., Frahm H., Göhmann F., Klümper A., Korepin V.E., The one-dimensional Hubbard model, Cambridge University Press, Cambridge, 2005.
  14. Izergin A.G., Korepin V.E., The quantum inverse scattering approach to correlation functions, Comm. Math. Phys. 94 (1984), 67-97.
  15. Izergin A.G., Korepin V.E., Reshetikhin N.Yu., Correlation functions in a one-dimensional Bose gas, J. Phys. A: Math. Gen. 20 (1987), 4799-4822.
  16. Ovchinnikov A.A., Coordinate space wave function from the algebraic Bethe ansatz for the inhomogeneous six-vertex model, Phys. Lett. A 374 (2010), 1311-1314, arXiv:1001.2672.
  17. Kitanine N., Correlation functions of the higher spin XXX chains, J. Phys. A: Math. Gen. 34 (2001), 8151-8169, math-ph/0104016.
  18. Castro-Alvaredo O.A., Maillet J.M., Form factors of integrable Heisenberg (higher) spin chains, J. Phys. A: Math. Theor. 40 (2007), 7451-7471, hep-th/0702186.
  19. Deguchi T., Matsui C., Form factors of integrable higher-spin XXZ chains and the affine quantum-group symmetry, Nuclear Phys. B 814 (2009), 405-438, arXiv:0807.1847.
    Deguchi T., Matsui C., Correlation functions of the integrable higher-spin XXX and XXZ chains through the fusion method, Nuclear Phys. B 831 (2010), 359-407, arXiv:0907.0582.
  20. Gaudin M., Bose gas in one dimension. I. The closure property of the scattering wavefunctions, J. Math. Phys. 12 (1971), 1674-1676.
    Gaudin M., Bose gas in one dimension. II. Orthogonality of the scattering states, J. Math. Phys. 12 (1971), 1677-1680.
  21. Fireman E.C., Lima-Santos A., Utiel W., Bethe ansatz solution for quantum spin-1 chains with boundary terms, Nuclear Phys. B 626 (2002), 435-462, nlin.SI/0110048.
  22. Melo C.S., Martins M.J., Algebraic Bethe ansatz for U(1) invariant integrable models: the method and general results, Nuclear Phys. B 806 (2009), 567-635, arXiv:0806.2404.
    Martins M.J., Melo C.S., Algebraic Bethe ansatz for U(1) invariant integrable models: compact and non-compact applications, Nuclear Phys. B 820 (2009), 620-648, arXiv:0902.3476.
  23. Belliard S., Ragoucy E., The nested Bethe ansatz for 'all' closed spin chains, J. Phys. A: Math. Theor. 41 (2008), 295202, 33 pages, arXiv:0804.2822.
  24. Crampé N., Ragoucy E., Simon D., Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions, J. Stat. Mech. Theory Exp. 2010 (2010), no. 11, P11038, 20 pages, arXiv:1009.4119.

Previous article   Next article   Contents of Volume 7 (2011)